I have connected my Wireless G Router #F5D7230-4 to the modem as instructed in the instructors manual and I am unable to obtain a internet connection. I have tried to disconnect from modem and router and reconnect several times but I am still unable to get a connection.

What kind of brand is it? and make sure your ethernet cable pluged in back of router where its supposed to be and reset your modem first when lights start lighting up reset your router...by reseting i mean unpluging power from equipment...hope that helps

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You are a bit confused with the various components required for Internet connections.
If you have a dial-up connection with an Internet Service Provider, all you do is connect your computer's modem port to the phone line and dial the phone number of your Internet Service provider.
If you are connected to the Internet Servise Provider by ADSL and have a combined ADSL router and modem then you connect the phone line to this router/modem. If the ADSL modem is separate component then the phone line is connected to the ADSL modem and this modem is connected to the router with a LAN cable and the computer is connected to the router with another LAN cable. If the route is wireless and the computer has a wireless card then you don't need a LAN cable between the router and the computer.
If you are connected to the Internet by ADSL, to secure your Internet connection and configure your computer so it can connect to the Internet, please follow my instruction - click on this link :-http://www.fixya.com/support/r3559128-wireless_router_security

4 steps to set up your home wireless network
1.Choose your wireless equipment
The first step is to make sure that you have the equipment you need. As you're looking for products in stores or on the Internet, you might notice that you can choose equipment that supports three different wireless networking technologies: 802.11a, 802.11b, and 802.11g. We recommend 802.11g, because it offers excellent performance and is compatible with almost everything.Shopping list
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Broadband Internet connection
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Wireless router
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A computer with built-in wireless networking support or a wireless network adapter

A wireless router
The router converts the signals coming across your Internet connection into a wireless broadcast, sort of like a cordless phone base station. Be sure to get a wireless router, and not a wireless access point.A wireless network adapter
Network adapters wirelessly connect your computer to your wireless router. If you have a newer computer you may already have wireless capabilities built in. If this is the case, then you will not need a wireless network adapter. If you need to purchase an adapter for a desktop computer, buy a USB wireless network adapter. If you have a laptop, buy a PC card-based network adapter. Make sure that you have one adapter for every computer on your network.Note: To make setup easy, choose a network adapter made by the same vendor that made your wireless router. For example, if you find a good price on a Linksys router, choose a Linksys network adapter to go with it. To make shopping even easier, buy a bundle, such as those available from D-Link, Netgear, Linksys, Microsoft, and Buffalo. If you have a desktop computer, make sure that you have an available USB port to plug the wireless network adapter into. If you don't have any open USB ports, buy a hub to add additional ports.
2.Connect your wireless router
Since you'll be temporarily disconnected from the Internet, print these instructions before you go any further.
First, locate your cable modem or DSL modem and unplug it to turn it off.
Next, connect your wireless router to your modem. Your modem should stay connected directly to the Internet. Later, after you've hooked everything up, your computer will wirelessly connect to your router, and the router will send communications through your modem to the Internet.
Next, connect your router to your modem:

Assuming your router is set correctly to your modem or the password and username are set correctly in a modem/router, make sure DHCP is enabled in the router, this gives each pc an ip address automatically.

Go to open internet explorer>tools>internet options>connections tab and ensure never dial a connection is ticked or greyed out. close IE

go to network connections, right click on your wireless adapter>properties>highlight TCP/IP line>properties and make sure that obtain an ip address automatically is ticked.

Open up internet explorer again and click refresh a few times, it will detect the proxy server then should connect, if not then the router configuration is wrong.

for dial up simply a modem .... on laptops if you do not have a builtin modem you can purchase a pc-card modem that goes into your pcmcia slot ... for highspeed either a ethernet (same option as modem) but almost all modern laptops have a built in ethernet ... or usb ... also I highly recommend a router if you are going highspeed ... even if you only have one computer ... a router will protect your system because it provides a NAT firewall. If you go with a router make sure to set up the internet connection on the router do not tunnel through the router with a PPPoE connection